By K Heiner Kamps, Timothy Porter
Summary homotopy conception is predicated at the statement that analogues of a lot of topological homotopy conception and straightforward homotopy idea exist in lots of different different types, reminiscent of areas over a set base, groupoids, chain complexes and module different types. learning specific types of homotopy constitution, resembling cylinders and course area structures permits not just a unified improvement of many examples of recognized homotopy theories, but in addition unearths the internal operating of the classical spatial conception, in actual fact indicating the logical interdependence of homes (in specific the life of convinced Kan fillers in linked cubical units) and effects (Puppe sequences, Vogt's lemma, Dold's Theorem on fibre homotopy equivalences, and homotopy coherence idea)
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Yu ; VI, ... ,"Cj, ... w(Y1 , ... a=1 j=l , Ya , ... 'yuH, 7f 1.. [Ya , Vj], VI, ... , Vj, ... , Vv ). A • A 36 4 BASIC FORMS, SPECTRAL SEQUENCE, CHARACTERISTIC FORM Finally for YI , ... 12) (d 2 ,-IW)(Yl , ... , Yu +2 ; VI"'" Vv - l ) L I (-l)U+"'+,6w(yl , ... , 17"" ... ,17,6, ... L[y"" Y,6], VI"'" Vv - I) 1~",<,6~u+2 The property d2 = 0 implies the relations 4-1 d6,1 = 0, = 0, dO,ldl,o + dl,odo,l = 0, dl ,od2 ,-1 + d2 ,-ldl ,o = 0, di,o + d 2 ,-ldo,1 + dO,ld2,-1 = O. 13) In the presence of a metric g this sequence is split, and an element W E E~'s is represented by w E W,S.
Another example is the Roussarie foliation on the unit tangent bundle of a Riemannian surface Mg (g > 1), considered in Chapter 2. It is a co dimension one foliation transverse to the circle fibers in T 1 M g ~ D2 xr 51, r = 7f1 (Mg). This foliation is definitely not Riemannian. This follows from the nontriviality of its Godbillon-Vey class. Namely a (transversally oriented) Riemannian foliation is in particular a 5 L( q)-foliation. This implies the vanishing of its Godbillon-Vey class. This class is constructed from a I-form 0: satisfying dl/ = 0: 1\ l/ for the transversal volume form l/ as the De Rham cohomology class [0: 1\ (do:P].
Chapter 4 Basic Forms, Spectral Sequence, Characteristic Form Basic forms Let F be an arbitrary foliation on a manifold M. A differential form w E W(M) is basic, if for all V E fL. i(V)w = 0, 8(V)w = 0 In a distinguished chart (Xl, ... ,Xp; Yl, ... : a, < ---l ---"'r = O. The exterior derivative preserves basic forms, since 8(V)dw = d8(V)w = 0, i(V)dw = 8(V)w di(V)w = 0 for w basic. Thus the set 0 11 == 0 11 (F) of all basic forms constitutes a sub complex d: OR - t O~+l of the De Rham complex O-(M).